# The Maclaurin’s series expansion of esin x is

By BYJU'S Exam Prep

Updated on: September 25th, 2023

(a) 1 + x – x2/2 + x4/12 – …..

(b) 1 – x +x2/2 – x4/8 + …..

(c) 1 + x + x2/2 – x4/8 + …..

(d) 1 + x + x2/2 – x4/12 + …..

The Maclaurin’s series expansion of esinx is 1 + x + x2/2 – x4/8 + …..

### Maclaurin’s Series Expansion of esin x

We know that,

• f(x) = esin x ⇒ f(0) = e = 1
• f'(x) = esin x (cos x) ⇒ f'(0) = 1
• f”(x) = esin x cos2 x + esin x (-sin x)
• f”(0) = 1 = esin x [cos2 x – sin x]
• f”'(x) = esin x [-sin 2x – cos x] + esin x [cos3 x – sin x cos x]
• f”'(x) = -1 + 1 = 0
• f (x) = esin x [-2 cos 2x + sin x] + esin x cos x [-sin 2x – cos x] + esin x [3 cos2 x(−sin x) − cos 2x/2 × 2] + esin x [cos4 x – sin x cos2 x]
• f (0) = -2 -1 -1 + 1 = -3

Substitue in Maclaurin Series

esin x = 1 + x + x2/2 + x3/3! (0) + x4/4! X (-3) + ……

= 1 + x + x2/2 – x4/8 + …..

Therefore, the Maclaurin’s series expansion of esin x is 1 + x + x2/2 – x4/8 + …..

Summary:

The Maclaurin’s series expansion of esin x is? (a) 1 + x – x2/2 + x4/12 – ….. (b) 1 – x +x2/2 – x4/8 + ….. (c) 1 + x + x2/2 – x4/8 + ….. (d) 1 + x + x2/2 – x4/12 + …..

1 + x + x2/2 – x4/8 + ….. is the Maclaurin’s series expansion of esinx. A Taylor series expansion of a function about 0 is called Maclaurin’s series. It is renamed after the Scottish mathematician Colin Maclaurin.

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